Most DBMS already do store metadata in the database. This data is general stored in what is broadly referred to as "system tables".
The metadata that a DBMS will store already will be what it needs to operate the database. Some of the types of metadata you've mentioned, e.g. ownership, permissions, and possibly even formats and descriptive names are already stored in your DBMS's system tables.
Other types of metadata (e.g. date/time of data entry) are record oriented, as opposed to the table or column oriented metadata that you can find in your DBMS's system tables. Record oriented metadata needs to be stored using a mechanism that you design and implement yourself. Typically, this means storing the record oriented metadata with your records directly. In some cases, you might find it helpful to normalize out some kinds of record oriented metadata into a common repository. An event log that tracks events for multiple tables is an example of such a common repository.
How you handle your metadata depends on what you need it for. As a matter of best practice, let the DBMS do what it is designed to do. Don't track metadata manually if the DBMS is doing it automatically. Only manually track the extra metadata.